Problem - Coronal Heating¶
The solar atmosphere is too hot compared to understood heating mechanisms.
What are the unknown heating mechanisms?
- Magnetic Reconnection
- Magnetohydrodynamic (MHD) Waves
- ...
Heating by MHD Waves¶
- Waves generated in high-energy convection zone.
- Waves propagate through the atmosphere, guided by magnetic field lines.
- Waves then have to deposit energy in the atmosphere.
The mechanism by which the wave energy is converted into atmospheric heating in the atmosphere is unknown, but the properties of the wave behaviour is dependent on the wave mode.
How are these waves generated and what are their properties?¶
Numerical Simulations of the lower solar atmosphere¶
The Code¶
The code used is the Sheffield Advanced Code (SAC) (Shelyag, Fedun, and Erdélyi 2008).
SAC simulates perturbations on a static background, using a CD4 solver with hyper-diffusion and hyper-viscosity terms to stabalise the solution.
This makes it well-suited to solving wave perturbations on top of a highly stratified background such as the solar atmosphere.
The Model¶
To simulate wave excitation in the photosphere a numerical model of the solar atmosphere is needed.
Hydrostatic background from the VAL III model C (Vernazza, Avrett, and Loeser 1981):
In [3]:
import pysac.mhs_atmosphere as atm
#Read in the VAL3C model
empirical_data = atm.hs_atmosphere.read_VAL3c_MTW(MTW_file=False)
# Create a Z array at the interpolated resolution and interpolate.
ZZ = u.Quantity(np.linspace(empirical_data['Z'][0], empirical_data['Z'][-1], 128), unit=empirical_data['Z'].unit)
table = atm.hs_atmosphere.interpolate_atmosphere(empirical_data, ZZ, s=0)
# Create a figure and make space for the axes.
fig, ax = plt.subplots(gridspec_kw={'right':0.77, 'left':0.16, 'bottom':0.13}, figsize=(13,8))
# Shortcut all the Mm conversion.
Z = empirical_data['Z'].to(u.Mm)
lrho, = ax.plot(Z, empirical_data['rho'].quantity.si, 'x', color='blue')
lrho_i, = ax.plot(ZZ.to(u.Mm), table['rho'].quantity.si, color='blue')
ax2 = ax.twinx()
lp, = ax2.plot(Z, empirical_data['p'].to(u.Pa), 'x', color='green')
lp_i, = ax2.plot(ZZ.to(u.Mm), table['p'].to(u.Pa), color='green')
ax3 = ax.twinx()
ax3.spines["right"].set_position(("axes", 1.2))
lt, = ax3.plot(Z, empirical_data['T'].to(u.K), 'x', color='red')
lt_i, = ax3.plot(ZZ.to(u.Mm), table['T'].to(u.K), color='red')
# Set primary axes properties and labels
ax.semilogy()
ax.set_ylabel(r"Density [{}]".format(lrho._yorig.unit._repr_latex_()))
ax.set_xlabel(r"Height [{}]".format(lrho._xorig.unit._repr_latex_()))
ax.set_xlim(Z[0].value-0.1, Z[-1].value+0.1)
# Pressure Axis
ax2.semilogy()
ax2.set_ylabel(r"Pressure [{}]".format(lp._yorig.unit._repr_latex_()))
# Temp axis
ax3.set_ylabel(r"Temperature [{}]".format(lt._yorig.unit._repr_latex_()))
ax.set_xlim([-0.02,1.62])
ax3.set_ylim([3500,7500])
# Set the colours for the ticks and the labels.
ax.tick_params(axis='y', colors=lrho.get_color())
ax2.tick_params(axis='y', colors=lp.get_color())
ax3.tick_params(axis='y', colors=lt.get_color())
ax.yaxis.label.set_color(lrho.get_color())
ax2.yaxis.label.set_color(lp.get_color())
ax3.yaxis.label.set_color(lt.get_color())
fig
Out[3]:
The Magnetic Flux Tube¶
The magnetic field model follows (Gent et al. 2013) and is constructed as a self-similar, cylindrically symmetric, expanding field.
In [5]:
slc = yt.SlicePlot(ds, 'x', 'density', origin='lower-center-domain', axes_unit='Mm')
slc.set_figure_size(10)
slc.set_cmap('all', 'viridis')
slc.set_font_size(20)
slc.set_zlim('all', 0, 2.5e-4)
seed_points = np.zeros([11,2]) + 1.52
seed_points[:,0] = np.linspace(-0.99, 0.95, seed_points.shape[0], endpoint=True)
slc.annotate_streamlines('mag_field_y', 'mag_field_z', field_color='magnetic_field_strength',
plot_args={'start_points':seed_points, 'density':15, 'cmap':'Blues', 'linewidth':2,
'norm': matplotlib.colors.LogNorm(*ds.all_data().quantities.extrema("magnetic_field_strength"))})
#force render
slc.save('/tmp/test.png')
#use the raw Figure for transparent bg
slc.plots['density'].figure
Out[5]:
Exciting Waves in the Photosphere¶
The dynamic photosphere with embedded magnetic field provides many potential ways of driving MHD waves.
- 'Buffeting' from convective motions
- Global oscillations
- Spiralling in downdrafts
Driving Waves in the Simulation Domain¶
$$ V(x,y,z) = F(x,y,z) \ e^{-\left(\frac{z^2}{\Delta z^2} + \frac{x^2}{\Delta x^2} + \frac{y^2}{\Delta y^2}\right)} \sin \left(2\pi \frac{t}{P}\right) $$
Identifying Waves from Broadband Drivers¶
Photospheric drivers excite multiple wave modes simulatenously.
How to quantify the relative strengths of the different modes from different drivers?
Assume uniform media:
- Locally decompose perturbations into Fast, Slow and Alfvén modes.
- Compare the percentage wave energy in each mode.
Relationship between Mode and Velocity Perturbation¶
Assuming parallel propagation (along the magnetic field) $k_\parallel >> k_\perp$ it can be shown that the slow mode and fast mode perturb different components of velocity:
Fast mode:
$$
\frac{|v_\parallel|}{|v_\perp|+|v_\parallel|} = \frac{\frac{k_\parallel}{k_\perp}|v_\perp|}{|v_\perp|+\frac{k_\parallel}{k_\perp}|v_\perp|}\\
= \frac{\frac{k_\parallel}{k_\perp}}{1+\frac{k_\parallel}{k_\perp}}\\
= \frac{1}{\frac{k_\perp}{k_\parallel}+1}\\
\ \\
|v_\parallel| >> |v_\perp|
$$
Slow mode:
$$
\frac{|v_\parallel|}{|v_\perp|+|v_\parallel|} = \frac{\frac{k_\perp}{k_\parallel}|v_\perp|}{|v_\perp|+\frac{k_\perp}{k_\parallel}|v_\perp|}\\
= \frac{\frac{k_\perp}{k_\parallel}}{1+\frac{k_\perp}{k_\parallel}}\\
= \frac{1}{\frac{k_\parallel}{k_\perp} + 1}\\
\ \\
|v_\parallel| << |v_\perp|
$$
Decomposing Velocity Pertubation in SAC¶
In [10]:
xc = domain['xmax']/2
yc = domain['ymax']/2
ti = 0
n = 100
surf_seeds = []
for theta in np.linspace(0, 2 * np.pi, n, endpoint=False):
surf_seeds.append([tube_r * np.cos(theta + 0.5 * ti) + xc,
tube_r * np.sin(theta + 0.5 * ti) + yc, domain['zmax']])
seeds = np.array(surf_seeds)
#Add axes:
axes, outline = mpf.add_axes(np.array(zip(ds.domain_left_edge,ds.domain_right_edge)).flatten()/1e8)
#Add seed points to plot:
seed_points = mlab.points3d(seeds[:,0], seeds[:,1], seeds[:,2],
color=(0.231, 0.298, 0.752), scale_mode='none',
scale_factor=1.5)
mlab_view(fig.scene)
Out[10]:
In [11]:
field_lines = SeedStreamline(seed_points = np.array(seeds))
bmag.add_child(field_lines)
field_lines.actor.mapper.scalar_visibility = False
field_lines.actor.property.color = (0,0,0)
field_lines.actor.property.line_width = 1.5
mlab_view(fig.scene)
Out[11]:
In [12]:
pd_seeds = ttf.make_circle_seeds(100, 60, **domain)
fieldlines, surface = ttf.create_flux_surface(bfield.outputs[0], pd_seeds)
surface.output.lines = None
flux_surface = mlab.pipeline.surface(surface.output)
flux_surface.actor.mapper.scalar_visibility = False
flux_surface.actor.property.color = (0.8,0.8,0.8)
#flux_surface.actor.property.line_width = 0
mlab_view(fig.scene)
Out[12]:
In [13]:
axes.visible = False
outline.visible = False
flux_surface.actor.property.edge_visibility = True
mlab_view(fig.scene, azimuth = 90, elevation = 75, distance=80, focalpoint=[63, 120, 110], aa=20)
Out[13]:
In [14]:
poly_norms = ttf.make_poly_norms(surface.output)
normvec = mlab.pipeline.glyph(poly_norms.output)
normvec.glyph.glyph_source.glyph_source = normvec.glyph.glyph_source.glyph_dict['arrow_source']
normvec.glyph.glyph.scale_mode = 'data_scaling_off'
normvec.glyph.glyph.color_mode = 'color_by_scale'
normvec.glyph.glyph.scale_factor = 5
normvec.glyph.glyph_source.glyph_position = 'tail'
mlab_view(fig.scene, azimuth=85, elevation=80, distance=50, focalpoint=[63, 120, 110], aa=20)
Out[14]:
MHD Waves excited by Different Photospheric Drivers¶
Chapter 4¶
Mumford, S. J., Fedun, V., Erdélyi, R. - The Astrophysical Journal - January 2015 - Volume 799, Issue 1
Generation of Magnetohydrodynamic Waves in Low Solar
Atmospheric Flux Tubes by Photospheric Motions
In [15]:
from streamlines import Streamlines
#Use Equation 1 to calculate the vector field in a 2D plane to plot it.
time = np.linspace(0,60,480)
dt = time[1:] - time [:-1]
period = 240.
x = np.linspace(7812.5,1992187.5,128)
y = np.linspace(7812.5,1992187.5,128)
x_max = x.max()
y_max = y.max()
xc = 1.0e6
yc = 1.0e6
xn = x - xc
yn = y - yc
delta_x=0.1e6
delta_y=0.1e6
xx, yy = np.meshgrid(xn,yn)
exp_y = np.exp(-(yn**2.0/delta_y**2.0))
exp_x = np.exp(-(xn**2.0/delta_x**2.0))
exp_x2, exp_y2= np.meshgrid(exp_x,exp_y)
exp_xyz = exp_x2 * exp_y2
#==============================================================================
# Define Driver Equations and Parameters
#==============================================================================
#A is the amplitude, B is the spiral expansion factor
A = 10
#Tdamp defines the damping of the driver with time, Tdep is the ocillator
tdamp = lambda time1: 1.0 #*np.exp(-(time1/(period)))
tdep = lambda time1: np.sin((time1*2.0*np.pi)/period) * tdamp(time1)
#Define a peak index to use for scaling in the inital frame
max_ind = np.argmax(tdep(time) > 0.9998)
def log():
B = 0.05
phi = np.arctan2(1,B)
theta = np.arctan2(yy,xx)
uy = np.sin(theta + phi)
ux = np.cos(theta + phi)
vx = lambda time1: (ux / np.sqrt(ux**2 + uy**2)) * exp_xyz * tdep(time1) * A
vy = lambda time1: (uy / np.sqrt(ux**2 + uy**2)) * exp_xyz * tdep(time1) * A
vv = np.sqrt(vx(time[max_ind])**2 + vy(time[max_ind])**2)
return vx, vy, vv
def arch():
B = 0.005
r = np.sqrt(xx**2 + yy**2)
vx = lambda time1: ( (B*1e6 * xx) / (xx**2 + yy**2) + yy/r ) * exp_xyz * tdep(time1) * A
vy = lambda time1: ( (B*1e6 * yy) / (xx**2 + yy**2) - xx/r ) * exp_xyz * tdep(time1) * A
vv = np.sqrt(vx(time[max_ind])**2 + vy(time[max_ind])**2)
return vx, vy, vv
def uniform():
#Uniform
vx = lambda time1: A * (yy / np.sqrt(xx**2 + yy**2)) * exp_xyz * tdep(time1)
vy = lambda time1: A * (-xx / np.sqrt(xx**2 + yy**2)) * exp_xyz * tdep(time1)
vv = np.sqrt(vx(time[max_ind])**2 + vy(time[max_ind])**2)
return vx, vy, vv
drivers = [log, arch, uniform]
#fig, axs = plt.subplots(1, 3, figsize=(18,9))
for driver_func in drivers:
fig, ax = plt.subplots(figsize=(7,6), dpi=300)
#============================================================================
# Do the Plotting
#============================================================================
vx, vy, vv = driver_func()
# Calculate Streamline
slines = Streamlines(x,y,vx(time[max_ind]),vy(time[max_ind]),maxLen=7000,
x0=xc, y0=yc, direction='forwards')
im = ax.imshow(vv, cmap='Blues', extent=[7812.5,x_max,7812.5,y_max])
im.set_norm(matplotlib.colors.Normalize(vmin=0,vmax=A))
#ax.hold()
if driver_func != uniform:
Sline, = ax.plot(slines.streamlines[0][0],slines.streamlines[0][1],color='red',linewidth=2, zorder=40)
else:
Sline = matplotlib.patches.Circle([1e6, 1e6], radius=.15e6, fill=False, color='red', linewidth=2, zorder=40)
ax.add_artist(Sline)
#Add colourbar
divider = make_axes_locatable(ax)
cax = divider.append_axes("right", size="5%", pad=0.2)
cbar = plt.colorbar(im,cax)
cbar.set_label(r"$|V|$ [ms$^{-1}$]")
scalar = matplotlib.ticker.ScalarFormatter(useMathText=False,useOffset=False)
scalar.set_powerlimits((-3,3))
cbar.formatter = scalar
cbar.ax.yaxis.get_offset_text().set_visible(True)
cbar.update_ticks()
#cbar.solids.set_rasterized(True)
cbar.solids.set_edgecolor("face")
#Add quiver plot overlay
qu = ax.quiver(x, y, vx(time[max_ind]), vy(time[max_ind]), scale=25*A, color='k', zorder=20, linewidth=1)
ax.axis([8.0e5,12.0e5,8.0e5,12.0e5])
ax.xaxis.set_major_formatter(scalar)
ax.yaxis.set_major_formatter(scalar)
ax.xaxis.set_major_locator(matplotlib.ticker.MaxNLocator(5))
ax.yaxis.set_major_locator(matplotlib.ticker.MaxNLocator(5))
ax.xaxis.get_offset_text().set_visible(False)
ax.yaxis.get_offset_text().set_visible(False)
ax.set_xlabel("X [Mm]")
ax.set_ylabel("Y [Mm]")
fig.savefig('images/driver_{}.png'.format(driver_func.__name__), transparent=True)
Archmedian Spiral¶
$$ F(x) = A \frac{B_Ax}{x^2 + y^2} \frac{y}{\sqrt{x^2 + y^2}},\\ F(y) = - A \frac{B_Ay}{x^2 + y^2} \frac{x}{\sqrt{x^2 + y^2}}. $$
Logarithmic Spiral¶
$$ F(x) = A \frac{\cos(\theta + \phi)}{\sqrt{x^2 + y^2}},\\ F(y) = - A \frac{\sin(\theta + \phi)}{\sqrt{x^2 + y^2}},\\ $$ where, $\theta = \tan^{-1}\left(\frac{y}{x}\right),\ \phi = \tan^{-1}\left(\frac{1}{B_L}\right)$
Analysis and Results¶
Wave Flux¶
Calculate wave energy flux from (Leroy 1985). $$ \vec{F}_{\textbf{wave}} \equiv \widetilde{p}_k \vec{v} + \frac{1}{\mu_0} \left(\vec{B}_b \cdot \vec{\widetilde{B}}\right) \vec{v} - \frac{1}{\mu_0}\left(\vec{v} \cdot \vec{\widetilde{B}} \right) \vec{B}_b $$
Time-Distance Diagrams¶
Vertical Driver¶
In [17]:
figures['vert']
Out[17]:
Horizontal Driver¶
In [18]:
figures['horiz']
Out[18]:
Logarithmic Spiral¶
In [19]:
figures['Slog']
Out[19]:
Comparison of Flux Averages¶
In [20]:
from flux_comparison import make_flux_bar_chart, get_averages
Favgs = get_averages(ch4.data_dir)
fig = make_flux_bar_chart((13,11), ch4.data_dir)
fig
Out[20]:
Effects of Logarithmic Spiral Expansion Factor¶
Chapter 5¶
Mumford, S. J. and Erdélyi, R. - Monthly Noticies of the Royal Astronomical Society - March 2015 - Volume 449 Issue 2.
Photospheric Logarithmic Velocity Spirals as MHD Wave Generation Mechanisms
Expansion Factor¶
$$
F(x) = A \frac{\cos(\theta + \phi)}{\sqrt{x^2 + y^2}},\\
F(y) = - A \frac{\sin(\theta + \phi)}{\sqrt{x^2 + y^2}},\\
$$
where,
$\theta = \tan^{-1}\left(\frac{y}{x}\right),\ \phi = \tan^{-1}\left(\frac{1}{B_L}\right)$
Using the result of (Bonet et al. 2008):
$$ B_L^{-1} = 6.4 \pm 1.6 = B_L = 0.15 $$
$$ B_L^{-1} = 6.4 \pm 1.6 = B_L = 0.15 $$
More Time-Distance Diagrams¶
$B=0.015$¶
In [24]:
bl_figures['B0015']
Out[24]:
$B=0.15$¶
In [25]:
bl_figures['B015']
Out[25]:
$B=1.5$¶
In [26]:
bl_figures['B15']
Out[26]:
Average Wave Flux¶
In [27]:
import os
int_periods = np.floor(600./cfg.period)*180
def calc_avgs(tube_r):
AvgsP = np.zeros([3,len(BL)])
for i, bl in enumerate(BL):
cfg.exp_fac = bl
times = np.load(os.path.join(cfg.data_dir, 'Times_{}.npy'.format(cfg.get_identifier())))
max_index = np.argmin(np.abs(int_periods - times))
Fpar, Fperp, Fphi = map(np.load, ph.glob_files(cfg, tube_r, 'LineFlux*Fp*npy'))
#Fpar, Fperp, Fphi = map(np.load, ph.glob_files(cfg, tube_r, '*vp*npy'))
Fpar[np.abs(Fpar)<1e-5], Fperp[np.abs(Fperp)<1e-5], Fphi[np.abs(Fphi)<1e-5] = 0., 0., 0.
Fpar, Fperp, Fphi = Fpar[:max_index,:], Fperp[:max_index,:], Fphi[:max_index,:]
Ftot2 = (Fpar**2 + Fperp**2 + Fphi**2)
Fpar2, Fperp2, Fphi2 = np.array([Fpar, Fperp, Fphi])**2
FparP, FperpP, FphiP = (Fpar2/Ftot2)*100, (Fperp2/Ftot2)*100, (Fphi2/Ftot2)*100
FparP = np.mean(np.ma.masked_array(FparP, np.isnan(FparP)))
FperpP = np.mean(np.ma.masked_array(FperpP, np.isnan(FperpP)))
FphiP = np.mean(np.ma.masked_array(FphiP, np.isnan(FphiP)))
AvgsP[:, i] = FparP, FperpP, FphiP
return AvgsP
figsize = (9.5,11)
fig, axs = plt.subplots(nrows=3, figsize=figsize, sharex=True)
titles = ["Flux Surface Radius $=156$ km", "Flux Surface Radius $=468$ km", "Flux Surface Radius $=936$ km"]
tubes = []
for i, ax in enumerate(axs):
AvgsP = calc_avgs(cfg.tube_radii[i])
tubes.append(AvgsP)
ax.semilogx()
ax.plot(BL, AvgsP[0], 'o', label=r"$F_\parallel^2$", mew=0, ms=10)
ax.plot(BL, AvgsP[1], '_', label=r"$F_\perp^2$", mew=2, ms=10)
ax.plot(BL, AvgsP[2], 'x', label=r"$F_\phi^2$", mew=2, ms=10)
ax.set_ylabel("% Square Wave Flux")
ax.set_title(titles[i])
ax.xaxis.set_major_formatter(matplotlib.ticker.ScalarFormatter())
ax.xaxis.set_ticks(BL)
ax.set_xlim([0.01, 2.01])
ax.set_ylim([0, 85])
err = np.array([-1*(0.15-1/(6.4-1.6)), 0.15+1/(6.4+1.6)])
ax.fill_betweenx(np.linspace(-5,105), err[0], err[1], alpha=0.3, color='green', linewidth=0)
axs[0].legend(loc=9)
axs[-1].set_xlabel("Logarithmic Spiral Expansion Factor ($B_L$)")
plt.tight_layout()
fig
Out[27]:
Maintaining Consistent Energy Input¶
The amplitude is calculated using:
$$
A^2 \propto \frac{1}{P}
$$
with $A=10$ $m/s$ and $P=240$ s as the reference point.
| Period [seconds] | Amplitude [ms$^{-1}$] |
| $30.0$ | $20\sqrt{2}$ |
| $60.0$ | $20$ |
| $90.0$ | $20\sqrt{\frac{2}{3}}$ |
| $120.0$ | $10\sqrt{2}$ |
| $150.0$ | $4\sqrt{10}$ |
| $180.0$ | $\frac{20}{\sqrt{3}}$ |
| $210.0$ | $20\sqrt{\frac{2}{7}}$ |
| $240.0$ | $10$ |
| $270.0$ | $\frac{20}{3}\sqrt{2}$ |
| $300.0$ | $4\sqrt{5}$ |
In [29]:
beta = False
cfg.exp_fac = 0.15
ph.xxlim = 600
tube_r = 'r30'
def add_all_bpert(ax, tube_r, N=4, levels=None):
kwargs = ph.get_triple(cfg, beta=beta, single='bpert')
x = kwargs['x_{}'.format(tube_r)]
y = kwargs['y_{}'.format(tube_r)]
par = kwargs['par_line_{}'.format(tube_r)].T[::-1, :]
par[np.abs(par)<=1e-12] = 0
perp = kwargs['perp_line_{}'.format(tube_r)].T[::-1, :]
perp[np.abs(perp)<=1e-12] = 0
phi = kwargs['phi_line_{}'.format(tube_r)].T[::-1, :]
phi[np.abs(phi)<=1e-12] = 0
ax[0].contour(x, y, par, N, colors='k', linewidths=np.linspace(0.5,1.5,N))
ax[1].contour(x, y, perp, N, colors='k', linewidths=np.linspace(0.5,1.5,N))
ax[2].contour(x, y, phi, N, colors='k', linewidths=np.linspace(0.5,1.5,N))
def add_triple_phase(ax, tube_r):
ps = ph.get_phase_speeds(cfg, tube_r)
for ax0 in ax:
ph.add_phase_speeds(ax0, color='g', **ps)
captions = {p: r"Period: ${}$ s amplitude: ".format(p) + a + r" ms$^{{-1}}$" for p, a in zip(periods, str_amps)[:10]}
#print(captions, file=sys.stderr)
width = 0.79
p_figures= {}
for i, paf in enumerate(all_periods):
[setattr(cfg, f, getattr(paf, f)) for f in paf._fields]
fig, ax = plt.subplots(nrows=3, ncols=2, figsize=(17,8), sharex=True)
kwargs = ph.get_single_velocity(cfg, 'r30', beta=beta)
kwargs.update(pvel_labels)
ph.triple_plot(ax[:,0], **kwargs)
kwargs = ph.get_single_percentage_flux(cfg, 'r30', beta=beta)
kwargs.update(pflux_labels)
kwargs.update({'cmap': 'PRGn'})
ph.triple_plot(ax[:,1], **kwargs)
#Remove the top two x labels
ax[0,0].set_xlabel('')
ax[1,0].set_xlabel('')
add_triple_phase(ax[:,0], 'r30')
ax[0,1].set_xlabel('')
ax[1,1].set_xlabel('')
add_triple_phase(ax[:,1], 'r30')
#add_all_bpert(ax, 'r30')
fig.tight_layout(h_pad=0.05)
p_figures[str(paf.period)] = fig
Even More Time-Distance Diagrams¶
$P=30$ $s$ $A=20\sqrt{2}$ $m/s$¶
In [30]:
p_figures['30.0']
Out[30]:
$P=150$ $s$ $A=4\sqrt{10}$ $m/s$¶
In [31]:
p_figures['150.0']
Out[31]:
$P=300$ $s$ $A=4\sqrt{5}$ $m/s$¶
In [32]:
p_figures['300.0']
Out[32]:
Period Comparison¶
In [33]:
from period_amps import periods, sim_params
sim_params = sim_params[:10]
periods = np.array(periods[:10])
fig, axs = plt.subplots(nrows=3, figsize=(10,11), sharex=True)
titles = ["Flux Surface Radius $=156$ km", "Flux Surface Radius $=468$ km", "Flux Surface Radius $=936$ km"]
tubes = []
for i, ax in enumerate(axs):
AvgsP = ph.get_all_avgs(cfg, cfg.tube_radii[i], sim_params)
tubes.append(AvgsP)
ax.plot(periods, AvgsP[0], 'o', label=r"$F_\parallel^2$", mew=0, ms=12)
ax.plot(periods, AvgsP[1], '_', label=r"$F_\perp^2$", mew=2, ms=12)
ax.plot(periods, AvgsP[2], 'x', label=r"$F_\phi^2$", mew=2, ms=12)
ax.set_ylabel("% Square Wave Flux")
ax.set_title(titles[i])
ax.xaxis.set_major_formatter(matplotlib.ticker.ScalarFormatter())
ax.xaxis.set_ticks(periods)
ax.set_xticklabels(["{}\n[{:n}]".format(p, 600//p) for p in periods])
ax.set_ylim([10, 75])
ax.set_xlim([25, 305])
axs[-1].set_xlabel("Period [s] \n [Number of periods averaged]")
axs[0].legend(loc=2)
#axs[0].legend(bbox_to_anchor=(1.06, 1.05))
plt.tight_layout(h_pad=0.1)
fig
Out[33]:
Conclusions¶
- Different spiral driver profiles do not have a large impact on the excited waves.
- Spirals with small expansion factors only excite $40-60$% Alfvén mode.
- Spirals with large expansion factors exicte mostly fast mode.
- Period only changes excited flux by $10-20$%
Bibliography¶
Bonet, J. A., I. Márquez, J. Sánchez Almeida, I. Cabello, and V. Domingo. 2008. “Convectively Driven Vortex Flows in the Sun.” The Astrophysical Journal 687 (2): L131–34. doi:10.1086/593329.
Gent, F. A., V. Fedun, and R. Erdélyi. 2014. “Magnetohydrostatic Equilibrium. II. Three-Dimensional Multiple Open Magnetic Flux Tubes in the Stratified Solar Atmosphere.” The Astrophysical Journal 789 (1): 42. doi:10.1088/0004-637X/789/1/42.
Gent, F. A., V. Fedun, S. J. Mumford, and R. Erdelyi. 2013. “Magnetohydrostatic Equilibrium - I. Three-Dimensional Open Magnetic Flux Tube in the Stratified Solar Atmosphere.” Monthly Notices of the Royal Astronomical Society 435 (1): 689–97. doi:10.1093/mnras/stt1328.
Leroy, Bernard. 1985. “On the Derivation of the Energy Flux of Linear Magnetohydrodynamic Waves.” Geophysical & Astrophysical Fluid Dynamics 32 (2): 123–33. doi:10.1080/03091928508208781.
Shelyag, S., V. Fedun, and R. Erdélyi. 2008. “Magnetohydrodynamic Code for Gravitationally-Stratified Media.” Astronomy and Astrophysics 486 (2): 655–62. doi:10.1051/0004-6361:200809800.
Vernazza, J. E., E. H. Avrett, and R. Loeser. 1981. “Structure of the Solar Chromosphere. III - Models of the EUV Brightness Components of the Quiet-Sun.” The Astrophysical Journal Supplement Series 45 (April): 635. doi:10.1086/190731.
My Publications¶
Mumford, S. J. and Erdélyi, R. - Monthly Noticies of the Royal Astronomical Society - March 2015 - Volume 449 Issue 2.
Photospheric Logarithmic Velocity Spirals as MHD Wave Generation Mechanisms
Mumford, S. J., Fedun, V., Erdélyi, R. - The Astrophysical Journal - January 2015 - Volume 799, Issue 1
Generation of Magnetohydrodynamic Waves in Low Solar
Atmospheric Flux Tubes by Photospheric Motions
The SunPy Community, Mumford, S. J., Christe, S., Pérez-Suárez, D., et. al - Computational Science and Discovery - January 2015 - Volume 8 Issue 1.
SunPy: Python for Solar Physics
Freij N., Scullion E. M., Nelson C. J., Mumford S. J., Wedemeyer S., and Erdélyi R. - The Astrophysical Journal - July 2014 - Volume 791, Issue 1, p.61
The Detection of Upwardly Propagating Waves Channeling Energy from the Chromosphere to the Low Corona
Gent, F. A., Fedun, V., Mumford, S. J., Erdélyi, R. - Monthly Notices of the Royal Astronomical Society - October 2013 - Volume 435, Issue 1, p.689-697
Magnetohydrostatic equilibrium - I. Three-dimensional open magnetic flux tube in the stratified solar atmosphere
Nelson, C. J., Doyle, J. G., Erdélyi, R., Huang, Z., Madjarska, M. S., Mathioudakis, M., Mumford, S. J., Reardon, K - Solar Physics - April 2013 - Volume 283, Issue 2, p.307-323.
Statistical Analysis of Small Ellerman Bomb Events